This paper presents a comprehensive algebraic analysis of the symmetry group of the cube, focusing on its rotation and spatial inversion symmetries. The study establishes that the full symmetry group of the cube is isomorphic to the direct product of the symmetric group on four elements and the cyclic group of order two, combining the symmetric group of degree four with the cyclic group of order two. We examine the group's structure through matrix representations, group multiplication tables, and detailed categorization of all elements by order. Further, the paper explores subgroup classifications, conjugacy classes, invariant subgroups, and the resulting cosets, quotient groups, and homomorphic correspondences. The Cayley graph of the octahedral group OhO_hOh is also constructed and analyzed, illustrating the connections between generator elements and group structure. A novel visualization using a heat map is introduced to represent the Cayley table, offering intuitive insight into group operations. This work provides a rigorous foundation for understanding the point groups of molecular structures that exhibit cubic or octahedral symmetry. The algebraic framework developed here enhances our comprehension of symmetry-dependent behaviors in molecular chemistry, atomic physics, and crystallography.